Numerical Models for Jump Diffusion Processes: A Finite Element Approach
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- Institutt for fysikk 
In this thesis we propose a finite element solution to the jump diffusion problem for pricing options. We focus on assets following CGMY processes and formulate the problem in weak bilinear form. In the case where the CGMY process is infinitely active there exist singular integrals in the formulation. Standard quadrature rules are therefore insufficient and we propose to solve these integrals exactly. This results in a finite element solution which handle assets with finite and infinite activity. An a priori analysis is included to show the method is stable and give an error estimate.The theory is extended to include American options by discussing two well known algorithms such as Brennan Schwartz and a penalization method. Finally the qualitative behavior of the solutions are illustrated both for the European and for the American case. We introduce pre calibrated market data as an attempt to show the variety of price evolution in the jump diffusion model compared to results following a standard Black Scholes model.